Vanishing Points and Horizon

Select two pair of parallel lines

We can do that by selecting the edges of a square

cd ~/Documents/Polimi/IACV/
I = imread('images/floor_tiles_hq.jpeg');
figure(1), imshow(I); figure(1), hold on;
[ptsx, ptsy] = getpts();
a = [ptsx(1); ptsy(1); 1];
b = [ptsx(2); ptsy(2); 1];
c = [ptsx(3); ptsy(3); 1];
d = [ptsx(4); ptsy(4); 1];

Let's now define the lines where the edges lie

lab = cross(a,b);
lbc = cross(b,c);
lcd = cross(c,d);
lad = cross(a,d);

Vanishing points

v1 = cross(lab, lcd);
v2 = cross(lbc, lad);

Horizon: line at infinity

h = cross(v1, v2);

Plot

figure(1), set(0, 'DefaultLineLineWidth', 2);

% show square

for p=[[a; b], [b; c], [c; d], [a; d]]

segment(p, 1);

end

figure(1), hold off;

% show points at infinity

figure(2), imshow(I); figure(2), hold on;

set(0, 'DefaultLineLineWidth', 2);

for vert = [a, b, c, d]

p1 = [vert; v1];

p2 = [vert; v2];

segment(p1, 2);

segment(p2, 2);

end

w1 = homogeneous_to_cartesian(v1);

w2 = homogeneous_to_cartesian(v2);

axis auto;

text(w1(1), w1(2), " v1", 'Fontsize', 20);

text(w2(1), w2(2), " v2", 'Fontsize', 20);

% show line at infinity

hc = homogeneous_to_cartesian_line(h);

x_coord = [min(w1(1), w2(1)), max(w1(1), w2(1))];

y_coord = hc(1) * x_coord + hc(2);

plot(x_coord, y_coord);

figure(2), hold off;

A plane has one horizon

Draw another pair of lines and check that they meet at the line at the same horizon.

Select the pair of lines.

figure(1), imshow(I); figure(1), hold on;
[ptsx, ptsy] = getpts();
e = [ptsx(1); ptsy(1); 1];
f = [ptsx(2); ptsy(2); 1];
g = [ptsx(3); ptsy(3); 1];
l = [ptsx(4); ptsy(4); 1];

Get the lines and find their intersection

lef = cross(e, f);
lgl = cross(g, l);
v3 = cross(lef, lgl);

Check if the point belongs to the line at infinity (it won't because of some imprecisions in the selection of the points)

h' * v3

ans = 2.2329e+18

axis auto;

x_coord = [0, 1500];

y_coord = hc(1) * x_coord + hc(2);

plot(x_coord, y_coord);

for vert = [e, f, g, l]

p1 = [vert; v3];

segment(p1, 1);

end

text(v3(1)/v3(3), v3(2)/v3(3), " v3", 'Fontsize', 20);

hold off;

function cart = homogeneous_to_cartesian(h)     % for points
    cart = [h(1)/h(3); h(2)/h(3)];
end
function cart = homogeneous_to_cartesian_line(h)    % for lines
    cart = [-h(1)/h(2); -h(3)/h(2)];
end
function segment(p, figure_number)              % draws a segment. p=[q1; q2] where qi are the points in hom coord
    p1 = homogeneous_to_cartesian(p(1:3));
    p2 = homogeneous_to_cartesian(p(4:6));
    figure(figure_number), plot([ p1(1), p2(1) ], [ p1(2), p2(2) ]);
end